Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising from optimal portfolio selection problem

TL;DR

This paper develops a monotone operator approach to solve a nonlinear Hamilton-Jacobi-Bellman equation from portfolio optimization, transforming it into a quasilinear form and proving existence and uniqueness of solutions.

Contribution

It introduces a novel application of the monotone operator method and Riccati transformation to analyze and solve the HJB equation in portfolio selection problems.

Findings

Established global Lipschitz continuity of the diffusion function.

Proved existence and uniqueness of solutions using Banach's fixed point theorem.

Applied the method to one-dimensional financial models.

Abstract

In this paper, we investigate a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach's fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space.